Final answer:
Hypothesis testing involves comparing the sample mean against the stated population mean to look for statistically significant differences, while constructing a confidence interval provides a range where the true mean is expected to be with a certain level of confidence. T-tests are used for hypothesis testing when the sample size is small and/or population standard deviation is unknown, while z-tests are used when the population standard deviation is known and sample size is large.
Step-by-step explanation:
Understanding Hypothesis Testing and Confidence Intervals
When we talk about hypothesis testing regarding the mean time students spend on homework, the null hypothesis (H0) generally represents the status quo, which in this case is H0: μ = 2.5 hours. The alternative hypothesis (H1) is that the mean time has increased, that is H1: μ > 2.5 hours. To determine whether we can reject the null hypothesis, we would typically use a t-test because although we know the population standard deviation (σ = 1.5), the sample size is relatively small (n = 26).
For constructing a confidence interval, such as finding a 95% confidence interval for pizza delivery times, we take the sample mean and add and subtract the margin of error to get the interval. The margin of error can be found by taking the critical value from the standard normal distribution (since σ is known and n >= 30 for the Central Limit Theorem to apply) and multiplying it by the standard error (σ/sqrt(n)).
In probability calculations, such as estimating the probability that the sample mean time to play a soccer match is between given hours, we convert the range to z-scores and use the standard normal distribution to find the associated probabilities.